On the Powers of a Matrix with Perturbations
On the Powers of a Matrix with Perturbations
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Date
2002-01-31
Authors
Stewart, G. W.
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Abstract
Let $A$ be a matrix of order $n$. The properties of the powers
$A^{k}$ of $A$ have been extensively studied in the literature.
This paper concerns the perturbed powers
\[
P_{k} = (A+E_{k})(A+E_{k-1})\cdots(A+E_{1}),
\]
where the $E_{k}$ are perturbation matrices. We will treat three
problems concerning the asymptotic behavior of the perturbed powers.
First, determine conditions under which $P_{k}\rightarrow 0$. Second,
determine the limiting structure of $P_{k}$. Third, investigate the
convergence of the power method with error: that is, given $u_{1}$,
determine the behavior of $u_{k} = \nu_{k}P_{k}u_{1}$, where $\nu_{k}$
is a suitable scaling factor.
(Also UMIACS-TR-2001-91)